Mathematics Consists of...
“Gauge fields are deeply related to some profoundly beautiful ideas of contemporary mathematics, ideas that are the driving forces of part of the mathematics of the last 40 years, . . . , the theory of fiber bundles.” Convinced that gauge fields are related to connections on fiber bundles, he tried to learn the fiber-bundle theory from several mathematical classics on the subject, but “learned nothing. The language of modern mathematics is too cold and abstract for a physicist”
The set X(M) of all C∞ vector fields on a manifold M has the structure of a real vector space, which is the same as a module over the field R of real numbers. Let F =C∞(M) again be the ring of C∞ functions on M. Since we can multiply a vector field by a C∞ function, the vector space X(M) is also a module over F. Thus the set X(M) has two module structures, over R and over F. In speaking of a linear map:
X(M) → X(M) one should be careful to specify whether it is R-linear or F-linear.
An F-linear map is of course R-linear, but the converse is not true.
Given an open subset U of a manifold M, one can think of U ×Rr as a family of vector spaces Rr parametrized by the points in U. A vector bundle, speaking, is a family of vector spaces that locally “looks” like U ×Rr.
Definition 7.1. A C∞ surjection π : E →M is a C∞ vector bundle of rank r if
(i) For every p ∈ M, the set Ep :=π−1(p) is a real vector space of dimension r;
(ii) every point p ∈ M has an open neighborhood U such that there is a fiberpreserving diffeomorphism
φU : π−1(U)→U ×Rr